On the volume functions and the cohomology rings of special weight varieties of type A

TL;DR

This paper explores the cohomology rings and symplectic volumes of certain type A weight varieties, establishing volume equivalences with flow polytopes and providing explicit cohomology presentations.

Contribution

It introduces a novel connection between symplectic volumes of weight varieties and flow polytope volumes, with explicit cohomology ring descriptions.

Findings

Symplectic volumes equal flow polytope volumes under certain conditions.

Explicit cohomology ring presentations for specific weight varieties.

Differential equations characterize volume functions of flow polytopes.

Abstract

In this paper, we consider the cohomology rings of some multiple weight varieties of type A, that is, symplectic torus quotients for a direct product of several coadjoint orbits of the special unitary group. Under some specific assumptions, we prove the symplectic volumes of multiple weight varieties are equal to the volumes of flow polytopes. Using differential equations satisfied by the volume functions of flow polytopes, we give an explicit presentation of the cohomology ring of the multiple weight variety of special type.